Volkswagen means exactly what it says: a car for the people. The name belongs to a political plan before it belonged to a brand.
Wikipedia says Ferdinand Porsche built a small, rear-engined car with an air-cooled engine in 1933, called the Volksauto. In 1934, Adolf Hitler asked for a basic family car at a target price of 990 Reichsmarks, paid for through a savings scheme.
The company that would build it was set up on May 28, 1937. The Volkswagen Group's own chronicle says three German Labour Front officials founded the Gesellschaft zur Vorbereitung des Deutschen Volkswagens, or "Gezuvor." That means a company to prepare the way for the German people's car. So the word was already in the title, but the firm had not yet built anything.
The car got a different name first. At the foundation-laying ceremony on May 26, 1938, the chronicle says Hitler named Porsche's vehicle the KdF-Wagen, after Kraft durch Freude, "Strength through Joy," the Labour Front's leisure organization. The savings scheme launched that August under the same name.
The company name changed within months. On September 16, 1938, the chronicle says, it became Volkswagenwerk GmbH. The car itself stayed the KdF-Wagen.
No saver received a car before 1945. The chronicle says 336,638 people had signed up by the end of the war. Only 630 KdF saloons were built.
The name Volkswagen became the car's name after the war. Deutschlandmuseum says the British military government renamed the production site Wolfsburg and the KdF-Wagen as Volkswagen. Under British control, the Volkswagen newsroom says, series production began on December 27, 1945, and 55 vehicles were built by the end of that year.